Finite Group Databasebeta

← #6508611

Subgroups — #6508611

Order 512.

3312 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic
2C22×#483C212yesmaximal, complemented
3C2×#2486C211yesmaximal, characteristic
4#24097C218yesmaximal, complemented
5#24063C214yesmaximal, complemented
6#24041C218yesmaximal, complemented
7#24003C214yesmaximal, complemented
8#21759C212yesmaximal, complemented
9#21738C211yesmaximal, characteristic
10#3501—216no—
11C2×#554—24no—
12C2×#554C2214yes—
13C2×D4×Q8C2214yes—
14C2×#544—21no—
15C2×#543—21no—
16C2×#543—21no—
17C2×#533C2212yes—
18C22×#123—21no—
19C22×C22⋊Q8C2212yes—
20C22×C22⋊Q8C2212yes—
21C2×#518—21no—
22C22×C4×D4C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C2×#483—216no—
25C2×#483C2218yes—
26C2×#483C2218yescomplemented
27C22×#106C2214yescomplemented
28C22×#103C2214yescomplemented
29C22×C22⋊C8C2212yes—
30#2486—24no—
31#2486C2218yes—
32C2×#398—21no—
33C2×#397—21no—
34C2×#389C2212yes—
35C2×#381—21no—
36C2×#380C2212yes—
37C2×#380C2211yescharacteristic
38#2058C2218yescomplemented
39#2043C2218yescomplemented
40#2035C22116yescomplemented
41#1892C2218yescomplemented
42#1872C2218yes—
43#1869C2218yes—
44#1865C22116yescomplemented
45#1853C2218yescomplemented
46#1824C2214yescomplemented
47#1817C2212yes—
48#1787C2218yescomplemented
49#1787C2214yescomplemented
50#1779C2212yes—
51C23×Q8C2312yes—
52C23×Q8C2312yes—
53C23×D4D411yescharacteristic
54C24×C4—21no—
55C24×C4C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C23×C8C2314yes—
58#554—264no—
59#554—216no—
60#554—216no—
61D4×Q8—216no—
62D4×Q8—216no—
63#544—216no—
64#543—216no—
65#543—216no—
66#533—216no—
67C2×C4⋊Q8—24no—
68C2×C4⋊Q8C2312yes—
69C2×C4⋊Q8C2312yes—
70C2×#125—24no—
71C2×#125—22no—
72C2×#123—28no—
73C2×#123—48no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8—28no—
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8—216no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8D4116yes—
86C2×C22⋊Q8C2314yes—
87C2×C22⋊Q8C2314yes—
88C2×C22⋊Q8C2314yes—
89#518—216no—
90C2×C4×Q8—22no—
91C2×C4×Q8C2314yes—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—28no—
95C2×C4×D4D414yes—
96C2×C4×D4C2314yes—
97C2×#117—22no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—22no—
103C22×C4⋊C4—22no—
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4—22no—
111C22×C22⋊C4—22no—
112C22×C22⋊C4—22no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4D411yescharacteristic
115C22×C22⋊C4D411yescharacteristic
116C22×C42C2311yescharacteristic
117#483—232no—
118#483—232no—
119C2×#106—216no—
120C2×#106C23116yes—
121C2×#103—216no—
122C2×#103C23116yes—
123C2×#103C23116yes—
124C2×C22⋊C8—28no—
125C2×C22⋊C8C2314yes—
126C2×C22⋊C8C2314yes—
127C2×C22⋊C8C2318yes—
128#398—28no—
129#397—28no—
130#389—24no—
131#389C2318yes—
132#381—28no—
133#380—24no—
134#380—22no—
135#380C2314yes—
136#380C2318yes—
137C2×#95—21no—
138C2×#95—22no—
139C2×#95—22no—
140#340C23116yes—
141#336C2318yes—
142C25C2×D411yescharacteristic
143C25C2×D411yescharacteristic
144C22×Q8—28no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—24no—
149C22×Q8—48no—
150C22×Q8—48no—
151C22×Q8C2×D418yes—
152C22×Q8C2414yes—
153C22×D4—24no—
154C22×D4—24no—
155C22×D4SD1618yes—
156C22×D4C2×D414yes—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D412yes—
176C23×C4C2×D418yes—
177C23×C4C2412yes—
178C23×C4C2411yescharacteristic
179C23×C4C2412yes—
180C22×C8—28no—
181C22×C8—48no—
182C22×C8C2418yes—
183C4⋊Q8—232no—
184C4⋊Q8—28no—
185C4⋊Q8—28no—
186#125—216no—
187#125—232no—
188#123—432no—
189C22⋊Q8—216no—
190C22⋊Q8—216no—
191C22⋊Q8—216no—
192C22⋊Q8—232no—
193C22⋊Q8—232no—
194C22⋊Q8—232no—
195C22⋊Q8—232no—
196C22⋊Q8—28no—
197C22⋊Q8—216no—
198C22⋊Q8—216no—
199C22⋊Q8—28no—
200C22⋊Q8—28no—
201C22⋊Q8—28no—
202C22⋊Q8C2×D4116yes—
203C22⋊Q8C2×D4116yes—
204C4×Q8—216no—
205C4×Q8—28no—
206C4×Q8—28no—
207C4×D4—232no—
208C4×D4—232no—
209C4×D4—216no—
210C4×D4—28no—
211C4×D4—28no—
212#117—216no—
213C2×C4⋊C4—216no—
214C2×C4⋊C4—216no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—48no—
237C2×C4⋊C4—48no—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×D418yes—
240C2×C4⋊C4C2414yes—
241C2×C4⋊C4C2414yes—
242C2×C4⋊C4C2414yes—
243C2×C22⋊C4—216no—
244C2×C22⋊C4—216no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—48no—
254C2×C22⋊C4C2×D414yes—
255C2×C22⋊C4C2×D414yes—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42C2×D412yes—
259C2×C42C2412yes—
260#106—232no—
261#103—232no—
262#103—416no—
263C22⋊C8—28no—
264C22⋊C8—28no—
265C22⋊C8—48no—
266#95—28no—
267#95—24no—
268#95—28no—
269C24—22no—
270C24—22no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24#12014yes—
276C24C2×SD1614yes—
277C24C2×SD1614yes—
278C24#13714yes—
279C24C22×D412yes—
280C24C22×D412yes—
281C24C22×D411yescharacteristic
282C2×Q8—28no—
283C2×Q8—28no—
284C2×Q8—28no—
285C2×Q8—24no—
286C2×Q8—216no—
287C2×Q8—416no—
288C2×Q8—84no—
289C2×Q8—84no—
290C2×Q8—84no—
291C2×Q8—88no—
292C2×Q8—816no—
293C2×Q8C22×D418yes—
294C2×D4—28no—
295C2×D4—28no—
296C2×D4—232no—
297C2×D4C2×SD16116yes—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—28no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—28no—
305C22×C4—28no—
306C22×C4—22no—
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—28no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—416no—
316C22×C4—44no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—48no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—48no—
328C22×C4—44no—
329C22×C4—48no—
330C22×C4—88no—
331C22×C4—84no—
332C22×C4—84no—
333C22×C4—84no—
334C22×C4—84no—
335C22×C4—816no—
336C22×C4#12014yes—
337C22×C4#12018yes—
338C22×C4C2×SD1614yes—
339C22×C4C22×D412yes—
340C22×C4C22×D412yes—
341C22×C4C22×D412yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D414yes—
344C22×C4C22×D414yes—
345C22×C4C22×D414yes—
346C22×C4C22×D412yes—
347C22×C4C2511yescharacteristic
348C2×C8—28no—
349C2×C8—216no—
350C2×C8—416no—
351C4⋊C4—28no—
352C4⋊C4—216no—
353C4⋊C4—28no—
354C4⋊C4—28no—
355C4⋊C4—216no—
356C4⋊C4—28no—
357C4⋊C4—216no—
358C4⋊C4—216no—
359C4⋊C4—28no—
360C4⋊C4—28no—
361C4⋊C4—24no—
362C4⋊C4—28no—
363C4⋊C4—28no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—48no—
367C4⋊C4—416no—
368C4⋊C4C22×D418yes—
369C22⋊C4—216no—
370C22⋊C4—216no—
371C22⋊C4—216no—
372C22⋊C4—28no—
373C22⋊C4—28no—
374C22⋊C4—48no—
375C22⋊C4—416no—
376C42—216no—
377C42—24no—
378C42—24no—
379C23—22no—
380C23—24no—
381C23—22no—
382C23—24no—
383C23—24no—
384C23—22no—
385C23—24no—
386C23—22no—
387C23—28no—
388C23—44no—
389C23—44no—
390C23—48no—
391C23—48no—
392C23—48no—
393C23—82no—
394C23—84no—
395C23—82no—
396C23#448116yes—
397C23C2×#12012yes—
398C23C2×#12014yes—
399C23C22×SD1614yes—
400C23C22×SD1614yes—
401C23C22×SD1612yes—
402C23C2×#13712yes—
403C23C2×#13714yes—
404C23C23×D411yescharacteristic
405Q8—216no—
406Q8—48no—
407Q8—88no—
408Q8—88no—
409Q8—88no—
410D4—232no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—28no—
415C2×C4—22no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—28no—
422C2×C4—28no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—48no—
426C2×C4—44no—
427C2×C4—48no—
428C2×C4—48no—
429C2×C4—48no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—84no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—88no—
440C2×C4—88no—
441C2×C4—88no—
442C2×C4—88no—
443C2×C4—88no—
444C2×C4—88no—
445C2×C4—816no—
446C2×C4C2×#12012yes—
447C2×C4C2×#12014yes—
448C2×C4C22×SD1614yes—
449C2×C4C23×D412yes—
450C2×C4#58514yes—
451C8—48no—
452C22—24no—
453C22—28no—
454C22—24no—
455C22—22no—
456C22—24no—
457C22—22no—
458C22—24no—
459C22—28no—
460C22—24no—
461C22—28no—
462C22—416no—
463C22—44no—
464C22—42no—
465C22—42no—
466C22—88no—
467C22C2×#44814yes—
468C22C2×#44818yes—
469C22C22×#12011yescharacteristic
470C22#348512yes—
471C22C23×SD1611yescharacteristic
472C22C22×#13711yescharacteristic
473C22#358612yes—
474C4—22no—
475C4—24no—
476C4—24no—
477C4—42no—
478C4—48no—
479C4—42no—
480C4—48no—
481C4—84no—
482C4—84no—
483C4#348514yes—
484C2—24no—
485C2—28no—
486C2—44no—
487C2#3544611yescharacteristic
488C2C22×#44811yescharacteristic
489C2#6037011yescharacteristic
490C2#6169512yes—
491C2#6170512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×#483C2semidirect
4. #24097C2semidirect
5. #24063C2semidirect
6. #24041C2semidirect
7. #24003C2semidirect
8. #21759C2semidirect
26. C2×#483C22semidirect
27. C22×#106C22semidirect
28. C22×#103C22semidirect
38. #2058C22semidirect
39. #2043C22semidirect
40. #2035C22semidirect
41. #1892C22semidirect
44. #1865C22semidirect
45. #1853C22semidirect
46. #1824C22semidirect
48. #1787C22semidirect
49. #1787C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#35012
C22×#4832
C2×#24862
#240972
#240632
#240412
#240032
#217592
#217382